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RSH_10_Ch_04 [Compatibility Mode]Introduction

Regression analysisRegression analysis is a very valuable tool for a manager

There are generally two purposes for regression analysis

4 – 2

regression analysis 1. To understand the relationship between

variables E.g. the relationship between the sales volume

and the advertising spending amount, the relationship between the price of a house and the square footage, etc.

2. To predict the value of one variable based on the value of another variable

Introduction

4 – 3

Multiple regression models have more than two variables

Nonlinear regression models are used when the relationships between the variables are not linear

Introduction

The variable to be predicted is called the dependent variabledependent variable Sometimes called the response variableresponse variable

The value of this variable depends on the value of the independent variableindependent variable

4 – 4

the value of the independent variableindependent variable Sometimes called the explanatoryexplanatory or

predictor variablepredictor variable

Independent variable1

Dependent variable

Independent variable2= + + ...

Prediction Relationship

Scatter Diagram

One way to investigate the relationship between variables is by plotting the data on a graph

Such a graph is often called a scatter scatter diagramdiagram or a scatter plotscatter plot

4 – 5

diagramdiagram or a scatter plotscatter plot The independent variable is normally

plotted on the X axis The dependent variable is normally

plotted on the Y axis

Triple A Construction renovates old homes They have found that the dollar volume of

renovation work each year is dependent on the area payroll

Triple A’s revenues and the total wage earnings for the past six years are listed below

Triple A Construction Example

TRIPLE A’S SALES ($100,000’s)

LOCAL PAYROLL ($100,000,000’s)

6 3 8 4 9 6 5 4 4.5 2 9.5 5Table 4.1

dependentdependent variablevariable

independentindependent variablevariable

4 – 7

Figure 4.1: Scatter Diagram for Triple A Constructi on Company Data in Table 4.1

6 –

4 –

2 –

0 –

| | | | | | | | 0 1 2 3 4 5 6 7 8

The graph indicates higher payroll seem to result in higher sales

A line has been drawn to show the relationship between the payroll and the sales

There is not a perfect relationship because not

Triple A Construction Example

4 – 8

There is not a perfect relationship because not all points lie in a straight line

Errors are involved if this line is used to predict sales based on payroll

Many lines could be drawn through these points, but which one best represents the true relationship ?

Simple Linear Regression

Regression models are used to find the relationship between variables – i.e. to predict the value of one variable based on the other

4 – 9

However there is some random error that cannot be predicted

Regression models can also be used to test if a relationship exists between variables

Simple Linear Regression

εεεεββββββββ ++++++++==== XY 10

where Y = dependent variable (response) X = independent variable (predictor or

explanatory) ββββ0 = intercept (value of Y when X = 0) ββββ1 = slope of the regression line εεεε = random error

Simple Linear Regression

The random error cannot be predicted. So an approximation of the model is used

XbbY 10 ++++====ˆ

Y = predicted value of Y X = independent variable (predictor or

explanatory) b0 = estimate of ββββ0

b1 = estimate of ββββ1

Triple A Construction

Triple A Construction is trying to predict sales based on area payroll

Y = Sales X = Area payroll

4 – 12

X = Area payroll

The line chosen in Figure 4.1 is the one that best fits the sample data by minimizing the sum of all errors

Error = (Actual value) – (Predicted value)

YYe ˆ−−−−====

Triple A Construction

The errors may be positive or negative – large positive and negative errors may cancel each other – result in very small average error – thus errors are squared

Error 2 = [(Actual value) – (Predicted value)] 2

4 – 13

22 )Y(Ye ˆ−−−−====

The best regression line is defined as the one that minimize the sum of squared errors, i.e. the total distance between the actual data points and the line

Triple A Construction

For the simple linear regression model, the values of the intercept and slope can be calculated from n sample data using the formulas below

XbbY 10 ++++====ˆ

X X ======== ∑∑∑∑

Y Y ======== ∑∑∑∑

Y X (X – X)2 (X – X)(Y – Y)

6 3 (3 – 4)2 = 1 (3 – 4)(6 – 7) = 1 8 4 (4 – 4)2 = 0 (4 – 4)(8 – 7) = 0

Regression calculations

4 – 15

8 4 (4 – 4)2 = 0 (4 – 4)(8 – 7) = 0 9 6 (6 – 4)2 = 4 (6 – 4)(9 – 7) = 4 5 4 (4 – 4)2 = 0 (4 – 4)(5 – 7) = 0 4.5 2 (2 – 4)2 = 4 (2 – 4)(4.5 – 7) = 5

9.5 5 (5 – 4)2 = 1 (5 – 4)(9.5 – 7) = 2.5

ΣY = 42 Y = 42/6 = 7

ΣX = 24 X = 24/6 = 4

Σ(X – X)2 = 10 Σ(X – X)(Y – Y) = 12.5

Table 4.2

4 – 17

Measuring the Fit of the Regression Model

Regression models can be developed for any variables X and Y

How do we know the model is good enough (with small errors) in predicting Y based on X ?

4 – 18

describing the accuracy of the model Three measures of variability

SST – Total variability about the mean SSE – Variability about the regression line SSR – Total variability that is explained by the

model

Sum of the squared error

∑∑∑∑ ∑∑∑∑ −−−−======== 22 )ˆ( YYeSSE

∑∑∑∑ −−−−==== 2)ˆ( YYSSR

Y X (Y – Y)2 Y (Y – Y)2 (Y – Y)2

^ ^^

9 6 (9 – 7)2 = 4 2 + 1.25(6) = 9.50 0.25 6.25

5 4 (5 – 7)2 = 4 2 + 1.25(4) = 7.00 4 0

4.5 2 (4.5 – 7)2 = 6.25 2 + 1.25(2) = 4.50 0 6.25

9.5 5 (9.5 – 7)2 = 6.25 2 + 1.25(5) = 8.25 1.5625 1.563

∑(Y – Y)2 = 22.5 ∑(Y – Y)2 = 6.875 ∑(Y – Y)2 = 15.625

Y = 7 SST = 22.5 SSE = 6.875 SSR = 15.625

^^

Measuring the Fit of the Regression Model

SST = 22.5 is the variability of the prediction using mean value of Y

SSE = 6.875 is the variability of the prediction using regression line

Prediction using regression line has reduced the

4 – 21

Prediction using regression line has reduced the variability by 22.5 −−−− 6.875 = 15.625

SSR = 15.625 indicates how much of the total variability in Y is explained by the regression model

Note: SST = SSR + SSE SSR – explained variability SSE – unexplained variability

Measuring the Fit of the Regression Model

12 –

10 –

8 –

4 – 22

Figure 4.2

Coefficient of Determination

The proportion of the variability in Y explained by regression equation is called the coefficient of coefficient of determinationdetermination

The coefficient of determination is r2

SSESSR r −−−−======== 12

625152 . .

. ========r

About 69% of the variability in Y is explained by the equation based on payroll ( X)

If SSE 0, then r 2 100%

Correlation Coefficient

It will always be between +1 and –1

2rr ±±±±====

4 – 24

It will always be between +1 and –1 Negative slope r < 0; positive slope r > 0 The correlation coefficient is r For Triple A Construction

8333069440 .. ========r

Correlation Coefficient

X

X

Y

4 – 26

Program 4.1A

4 – 27

Program 4.1B

4 – 28

Program 4.1C

Correlation coefficient ( r) is Multiple R in Excel

4 – 29

1. Errors are independent 2. Errors are normally distributed

If we make certain assumptions about the errors in a regression model, we can perform statistical tests to determine if the model is useful

4 – 30

2. Errors are normally distributed 3. Errors have a mean of zero 4. Errors have a constant variance

A plot of the residuals (errors) will often highlig ht any glaring violations of the assumption

Residual Plots

4 – 31

Figure 4.4A

E rr

Nonconstant error variance – violation Errors increase as X increases, violating the

constant variance assumption

decreasing indicate that the model is not linear (perhaps quadratic)

4 – 33

Figure 4.4C

E rr

Estimating the Variance

Errors are assumed to have a constant variance ( σσσσ 2), but we usually don’t know this

It can be estimated using the mean mean squared errorsquared error (MSEMSE), s2

4 – 34

1 2

MSEs

where n = number of observations in the sample k = number of independent variables

Estimating the Variance

4 – 35

We can estimate the standard deviation, s This is also called the standard error of the standard error of the

estimateestimate or the standard deviation of the standard deviation of the regressionregression

31171881 .. ============ MSEs

A small s2 or s means the actual data deviate within a small range from the predicted result

Testing the Model for Significance

Both r2 and the MSE (s2) provide a measure of accuracy in a regression model

However when the sample size is too small, you can get good values for MSE and r2

even if there is no relationship between the

4 – 36

even if there is no relationship between the variables

Testing the model for significance helps determine if r2 and MSE are meaningful and if a linear relationship exists between the variables

We do this by performing a statistical hypothesis test

Testing the Model for Significance

We start with the general linear model

εεεεββββββββ ++++++++==== XY 10

If ββββ1 = 0, the null hypothesis is that there is nono relationship between X and Y

4 – 37

nono relationship between X and Y The alternate hypothesis is that there isis a

linear relationship ( ββββ1 ≠ 0) If the null hypothesis can be rejected, we

have proven there is a linear relationship We use the F statistic for this test

A continuous probability distribution (Fig. 2.15) The area underneath the curve represents

probability of the F statistic value falling within a particular interval.

The F statistic is the ratio of two sample variances

The F Distribution

4 – 38

The F statistic is the ratio of two sample variances F distributions have two sets of degrees of

freedom Degrees of freedom are based on sample size and

used to calculate the numerator and denominator

df1 = degrees of freedom for the numerator df2 = degrees of freedom for the denominator

The F Distribution

Consider the example:

4 – 39

Fαααα, df1, df2 = F0.05, 5, 6 = 4.39

This means

P(F > 4.39) = 0.05

There is only a 5% probability that F will exceed 4.39 (see Fig. 2.16)

The F Distribution

The F Distribution

F value for 0.05 probability with 5 and 6 degrees of freedom

4 – 41

Figure 2.16

F = 4.39

Testing the Model for Significance

The F statistic for testing the model is based on the MSE (s2) and mean squared regression ( MSR)

k SSR

MSR ==== where

4 – 42

The F statistic is

F ====

This describes an F distribution with degrees of freedom for the numerator = df1 = k degrees of freedom for the denominator = df2 = n – k – 1

Testing the Model for Significance

If there is very little error, the MSE would be small and the F-statistic would be large indicating the model is useful

If the F-statistic is large, the significance level ( p-value) will be low, indicating it is

4 – 43

level ( p-value) will be low, indicating it is unlikely this would have occurred by chance

So when the F-value is large, we can reject the null hypothesis and accept that there is a linear relationship between X and Y and the values of the MSE and r2 are meaningful

Steps in a Hypothesis Test

1. Specify null and alternative hypotheses 010 ====ββββ:H 011 ≠≠≠≠ββββ:H

2. Select the level of significance ( αααα). Common

4 – 44

2. Select the level of significance ( αααα). Common values are 0.01 and 0.05

3. Calculate the value of the test statistic using the formula

MSE MSR

Steps in a Hypothesis Test

4. Make a decision using one of the following methods a) Reject the null hypothesis if the test statistic is

greater than the FF--valuevalue from the table in Appendix D. Otherwise, do not reject the null hypothesis:

4 – 45

kdf ====1

12 −−−−−−−−==== kndf

b) Reject the null hypothesis if the observed signific ance level, or pp--value,value, is less than the level of significance (αααα). Otherwise, do not reject the null hypothesis:

)( statistictest calculatedvalue- >>>>==== FPp αααα<<<<value- ifReject p

Triple A Construction Step 1.Step 1.

H0: ββββ1 = 0 (no linear relationship between X and Y)

H1: ββββ1 ≠ 0 (linear relationship exists between X and Y)

Step 2.Step 2.

625015 1 625015

Step 3.Step 3. Calculate the value of the test statistic

Triple A Construction Step 4.Step 4.

Reject the null hypothesis if the test statistic is greater than the F-value in Appendix D

df1 = k = 1 df2 = n – k – 1 = 6 – 1 – 1 = 4

4 – 47

df2 = n – k – 1 = 6 – 1 – 1 = 4

The value of F associated with a 5% level of significance and with degrees of freedom 1 and 4 is found in Appendix D

F0.05,1,4 = 7.71 Fcalculated = 9.09 Reject H0 because 9.09 > 7.71

Triple A Construction

We can conclude there is a statistically significant relationship between X and Y

The r2 value of 0.69 means about 69% of the variability in

4 – 48

F = 7.71

9.09Figure 4.5

about 69% of the variability in sales ( Y) is explained by local payroll ( X)

Triple A Construction

The F-test determines whether or not there is a relationship between the variables

r2 (coefficient of determination) is the best measure of the strength of the prediction

4 – 49

measure of the strength of the prediction relationship between the X and Y variables Values closer to 1 indicate a strong prediction

relationship Good regression models have a low

significance level for the F-test and high r2

value.

Analysis of Variance (ANOVA) Table

When software is used to develop a regression model, an ANOVA table is typically created that shows the observed significance level ( p-value) for the calculated F value

This can be compared to the level of significance (αααα) to make a decision

4 – 50

Regression k SSR MSR = SSR/k MSR/MSE P(F > MSR/MSE)

Residual n - k - 1 SSE MSE = SSE/(n - k - 1)

Total n - 1 SST

4 – 51

Because this probability is less than 0.05, we reject the null hypothesis of no linear relationshi p and conclude there is a linear relationship between X and Y

Program 4.1D (partial)

P(F > 9.0909) = 0.0394

Multiple Regression Analysis

Multiple regression modelsMultiple regression models are extensions to the simple linear model and allow the creation of models with several independent variables

4 – 52

Y = dependent variable (response variable) Xi = ith independent variable (predictor or explanatory

variable) ββββ0 = intercept (value of Y when all Xi = 0) ββββI = coefficient of the ith independent variable k = number of independent variables εεεε = random error

Multiple Regression Analysis

To estimate these values, samples are taken and the following equation is developed

kk XbXbXbbY ++++++++++++++++==== ...ˆ 22110

where = predicted value of Y

b0 = sample intercept (and is an estimate of ββββ0) bi = sample coefficient of the ith variable (and is

an estimate of ββββi)

Jenny Wilson Realty

Jenny Wilson wants to develop a model to determine the suggested listing price for houses based on the size and age of the house

22110 ˆ XbXbbY ++++++++====

where = predicted value of dependent variable (selling pri ce)

b0 = Y intercept X1 and X2 = value of the two independent variables (square

footage and age) respectively b1 and b2 = slopes for X1 and X2 respectively

Y

She selects a few samples of the houses sold recently and records the data shown in Table 4.5

She also saves information on house condition to be used later

Jenny Wilson Realty SELLING PRICE ($)

SQUARE FOOTAGE AGE CONDITION

95,000 1,926 30 Good

119,000 2,069 40 Excellent

124,800 1,720 30 Excellent

135,000 1,396 15 Good

142,000 1,706 32 Mint

Jenny Wilson Realty

0021788.0

Evaluating Multiple Regression Models

Evaluation is similar to simple linear regression models The p-value for the F-test and r2 are

interpreted the same

4 – 57

The hypothesis is different because there is more than one independent variable The F-test is investigating whether all

the coefficients are equal to 0 If the F-test is significant, it does not

mean all independent variables are significant

Evaluating Multiple Regression Models

To determine which independent variables are significant, tests are performed for each variable

010 ====ββββ:H

4 – 58

010 ====ββββ:H 011 ≠≠≠≠ββββ:H

The test statistic is calculated and if the p-value is lower than the level of significance ( αααα), the null hypothesis is rejected

Jenny Wilson Realty

The model is statistically significant The p-value for the F-test is 0.002 r2 = 0.6719 so the model explains about 67% of

the variation in selling price ( Y) But the F-test is for the entire model and we can’t

4 – 59

But the F-test is for the entire model and we can’t tell if one or both of the independent variables ar e significant

By calculating the p-value of each variable, we can assess the significance of the individual variables

Since the p-value for X1 (square footage) and X2 (age) are both less than the significance level of 0.05, both null hypotheses can be rejected

Binary or Dummy Variables

BinaryBinary (or dummydummy or indicatorindicator) variables are special variables created for qualitative data

A binary variable is assigned a value of 1 if a particular qualitative condition is met and

4 – 60

a particular qualitative condition is met and a value of 0 otherwise

Adding binary variables may increase the accuracy of the regression model

The number of binary variables must be one less than the number of categories of the qualitative variable

Jenny Wilson Realty

Jenny believes a better model can be developed if she includes information about the condition of the property

X3 = 1 if house is in excellent condition = 0 otherwise

4 – 61

= 0 otherwise X4 = 1 if house is in mint (perfect) condition

= 0 otherwise

Two binary variables are used to describe the three categories of condition

No variable is needed for “good” condition since if both X3 = 0 and X4 = 0, the house must be in good condition

Jenny Wilson Realty

Model explains about 89.8% of the variation in selling price

F-value

4 – 64

Program 4.3 – The two additional dummy variables result in higher r2 and smaller significance value.

F-value indicates significance

Model Building

The best model is a statistically significant model with a high r2 and few variables

As more variables are added to the model, the r2-value usually increases However more variables does not

4 – 65

However more variables does not necessarily mean better model

For this reason, the adjusted adjusted rr22 value…

Regression analysisRegression analysis is a very valuable tool for a manager

There are generally two purposes for regression analysis

4 – 2

regression analysis 1. To understand the relationship between

variables E.g. the relationship between the sales volume

and the advertising spending amount, the relationship between the price of a house and the square footage, etc.

2. To predict the value of one variable based on the value of another variable

Introduction

4 – 3

Multiple regression models have more than two variables

Nonlinear regression models are used when the relationships between the variables are not linear

Introduction

The variable to be predicted is called the dependent variabledependent variable Sometimes called the response variableresponse variable

The value of this variable depends on the value of the independent variableindependent variable

4 – 4

the value of the independent variableindependent variable Sometimes called the explanatoryexplanatory or

predictor variablepredictor variable

Independent variable1

Dependent variable

Independent variable2= + + ...

Prediction Relationship

Scatter Diagram

One way to investigate the relationship between variables is by plotting the data on a graph

Such a graph is often called a scatter scatter diagramdiagram or a scatter plotscatter plot

4 – 5

diagramdiagram or a scatter plotscatter plot The independent variable is normally

plotted on the X axis The dependent variable is normally

plotted on the Y axis

Triple A Construction renovates old homes They have found that the dollar volume of

renovation work each year is dependent on the area payroll

Triple A’s revenues and the total wage earnings for the past six years are listed below

Triple A Construction Example

TRIPLE A’S SALES ($100,000’s)

LOCAL PAYROLL ($100,000,000’s)

6 3 8 4 9 6 5 4 4.5 2 9.5 5Table 4.1

dependentdependent variablevariable

independentindependent variablevariable

4 – 7

Figure 4.1: Scatter Diagram for Triple A Constructi on Company Data in Table 4.1

6 –

4 –

2 –

0 –

| | | | | | | | 0 1 2 3 4 5 6 7 8

The graph indicates higher payroll seem to result in higher sales

A line has been drawn to show the relationship between the payroll and the sales

There is not a perfect relationship because not

Triple A Construction Example

4 – 8

There is not a perfect relationship because not all points lie in a straight line

Errors are involved if this line is used to predict sales based on payroll

Many lines could be drawn through these points, but which one best represents the true relationship ?

Simple Linear Regression

Regression models are used to find the relationship between variables – i.e. to predict the value of one variable based on the other

4 – 9

However there is some random error that cannot be predicted

Regression models can also be used to test if a relationship exists between variables

Simple Linear Regression

εεεεββββββββ ++++++++==== XY 10

where Y = dependent variable (response) X = independent variable (predictor or

explanatory) ββββ0 = intercept (value of Y when X = 0) ββββ1 = slope of the regression line εεεε = random error

Simple Linear Regression

The random error cannot be predicted. So an approximation of the model is used

XbbY 10 ++++====ˆ

Y = predicted value of Y X = independent variable (predictor or

explanatory) b0 = estimate of ββββ0

b1 = estimate of ββββ1

Triple A Construction

Triple A Construction is trying to predict sales based on area payroll

Y = Sales X = Area payroll

4 – 12

X = Area payroll

The line chosen in Figure 4.1 is the one that best fits the sample data by minimizing the sum of all errors

Error = (Actual value) – (Predicted value)

YYe ˆ−−−−====

Triple A Construction

The errors may be positive or negative – large positive and negative errors may cancel each other – result in very small average error – thus errors are squared

Error 2 = [(Actual value) – (Predicted value)] 2

4 – 13

22 )Y(Ye ˆ−−−−====

The best regression line is defined as the one that minimize the sum of squared errors, i.e. the total distance between the actual data points and the line

Triple A Construction

For the simple linear regression model, the values of the intercept and slope can be calculated from n sample data using the formulas below

XbbY 10 ++++====ˆ

X X ======== ∑∑∑∑

Y Y ======== ∑∑∑∑

Y X (X – X)2 (X – X)(Y – Y)

6 3 (3 – 4)2 = 1 (3 – 4)(6 – 7) = 1 8 4 (4 – 4)2 = 0 (4 – 4)(8 – 7) = 0

Regression calculations

4 – 15

8 4 (4 – 4)2 = 0 (4 – 4)(8 – 7) = 0 9 6 (6 – 4)2 = 4 (6 – 4)(9 – 7) = 4 5 4 (4 – 4)2 = 0 (4 – 4)(5 – 7) = 0 4.5 2 (2 – 4)2 = 4 (2 – 4)(4.5 – 7) = 5

9.5 5 (5 – 4)2 = 1 (5 – 4)(9.5 – 7) = 2.5

ΣY = 42 Y = 42/6 = 7

ΣX = 24 X = 24/6 = 4

Σ(X – X)2 = 10 Σ(X – X)(Y – Y) = 12.5

Table 4.2

4 – 17

Measuring the Fit of the Regression Model

Regression models can be developed for any variables X and Y

How do we know the model is good enough (with small errors) in predicting Y based on X ?

4 – 18

describing the accuracy of the model Three measures of variability

SST – Total variability about the mean SSE – Variability about the regression line SSR – Total variability that is explained by the

model

Sum of the squared error

∑∑∑∑ ∑∑∑∑ −−−−======== 22 )ˆ( YYeSSE

∑∑∑∑ −−−−==== 2)ˆ( YYSSR

Y X (Y – Y)2 Y (Y – Y)2 (Y – Y)2

^ ^^

9 6 (9 – 7)2 = 4 2 + 1.25(6) = 9.50 0.25 6.25

5 4 (5 – 7)2 = 4 2 + 1.25(4) = 7.00 4 0

4.5 2 (4.5 – 7)2 = 6.25 2 + 1.25(2) = 4.50 0 6.25

9.5 5 (9.5 – 7)2 = 6.25 2 + 1.25(5) = 8.25 1.5625 1.563

∑(Y – Y)2 = 22.5 ∑(Y – Y)2 = 6.875 ∑(Y – Y)2 = 15.625

Y = 7 SST = 22.5 SSE = 6.875 SSR = 15.625

^^

Measuring the Fit of the Regression Model

SST = 22.5 is the variability of the prediction using mean value of Y

SSE = 6.875 is the variability of the prediction using regression line

Prediction using regression line has reduced the

4 – 21

Prediction using regression line has reduced the variability by 22.5 −−−− 6.875 = 15.625

SSR = 15.625 indicates how much of the total variability in Y is explained by the regression model

Note: SST = SSR + SSE SSR – explained variability SSE – unexplained variability

Measuring the Fit of the Regression Model

12 –

10 –

8 –

4 – 22

Figure 4.2

Coefficient of Determination

The proportion of the variability in Y explained by regression equation is called the coefficient of coefficient of determinationdetermination

The coefficient of determination is r2

SSESSR r −−−−======== 12

625152 . .

. ========r

About 69% of the variability in Y is explained by the equation based on payroll ( X)

If SSE 0, then r 2 100%

Correlation Coefficient

It will always be between +1 and –1

2rr ±±±±====

4 – 24

It will always be between +1 and –1 Negative slope r < 0; positive slope r > 0 The correlation coefficient is r For Triple A Construction

8333069440 .. ========r

Correlation Coefficient

X

X

Y

4 – 26

Program 4.1A

4 – 27

Program 4.1B

4 – 28

Program 4.1C

Correlation coefficient ( r) is Multiple R in Excel

4 – 29

1. Errors are independent 2. Errors are normally distributed

If we make certain assumptions about the errors in a regression model, we can perform statistical tests to determine if the model is useful

4 – 30

2. Errors are normally distributed 3. Errors have a mean of zero 4. Errors have a constant variance

A plot of the residuals (errors) will often highlig ht any glaring violations of the assumption

Residual Plots

4 – 31

Figure 4.4A

E rr

Nonconstant error variance – violation Errors increase as X increases, violating the

constant variance assumption

decreasing indicate that the model is not linear (perhaps quadratic)

4 – 33

Figure 4.4C

E rr

Estimating the Variance

Errors are assumed to have a constant variance ( σσσσ 2), but we usually don’t know this

It can be estimated using the mean mean squared errorsquared error (MSEMSE), s2

4 – 34

1 2

MSEs

where n = number of observations in the sample k = number of independent variables

Estimating the Variance

4 – 35

We can estimate the standard deviation, s This is also called the standard error of the standard error of the

estimateestimate or the standard deviation of the standard deviation of the regressionregression

31171881 .. ============ MSEs

A small s2 or s means the actual data deviate within a small range from the predicted result

Testing the Model for Significance

Both r2 and the MSE (s2) provide a measure of accuracy in a regression model

However when the sample size is too small, you can get good values for MSE and r2

even if there is no relationship between the

4 – 36

even if there is no relationship between the variables

Testing the model for significance helps determine if r2 and MSE are meaningful and if a linear relationship exists between the variables

We do this by performing a statistical hypothesis test

Testing the Model for Significance

We start with the general linear model

εεεεββββββββ ++++++++==== XY 10

If ββββ1 = 0, the null hypothesis is that there is nono relationship between X and Y

4 – 37

nono relationship between X and Y The alternate hypothesis is that there isis a

linear relationship ( ββββ1 ≠ 0) If the null hypothesis can be rejected, we

have proven there is a linear relationship We use the F statistic for this test

A continuous probability distribution (Fig. 2.15) The area underneath the curve represents

probability of the F statistic value falling within a particular interval.

The F statistic is the ratio of two sample variances

The F Distribution

4 – 38

The F statistic is the ratio of two sample variances F distributions have two sets of degrees of

freedom Degrees of freedom are based on sample size and

used to calculate the numerator and denominator

df1 = degrees of freedom for the numerator df2 = degrees of freedom for the denominator

The F Distribution

Consider the example:

4 – 39

Fαααα, df1, df2 = F0.05, 5, 6 = 4.39

This means

P(F > 4.39) = 0.05

There is only a 5% probability that F will exceed 4.39 (see Fig. 2.16)

The F Distribution

The F Distribution

F value for 0.05 probability with 5 and 6 degrees of freedom

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Figure 2.16

F = 4.39

Testing the Model for Significance

The F statistic for testing the model is based on the MSE (s2) and mean squared regression ( MSR)

k SSR

MSR ==== where

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The F statistic is

F ====

This describes an F distribution with degrees of freedom for the numerator = df1 = k degrees of freedom for the denominator = df2 = n – k – 1

Testing the Model for Significance

If there is very little error, the MSE would be small and the F-statistic would be large indicating the model is useful

If the F-statistic is large, the significance level ( p-value) will be low, indicating it is

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level ( p-value) will be low, indicating it is unlikely this would have occurred by chance

So when the F-value is large, we can reject the null hypothesis and accept that there is a linear relationship between X and Y and the values of the MSE and r2 are meaningful

Steps in a Hypothesis Test

1. Specify null and alternative hypotheses 010 ====ββββ:H 011 ≠≠≠≠ββββ:H

2. Select the level of significance ( αααα). Common

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2. Select the level of significance ( αααα). Common values are 0.01 and 0.05

3. Calculate the value of the test statistic using the formula

MSE MSR

Steps in a Hypothesis Test

4. Make a decision using one of the following methods a) Reject the null hypothesis if the test statistic is

greater than the FF--valuevalue from the table in Appendix D. Otherwise, do not reject the null hypothesis:

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kdf ====1

12 −−−−−−−−==== kndf

b) Reject the null hypothesis if the observed signific ance level, or pp--value,value, is less than the level of significance (αααα). Otherwise, do not reject the null hypothesis:

)( statistictest calculatedvalue- >>>>==== FPp αααα<<<<value- ifReject p

Triple A Construction Step 1.Step 1.

H0: ββββ1 = 0 (no linear relationship between X and Y)

H1: ββββ1 ≠ 0 (linear relationship exists between X and Y)

Step 2.Step 2.

625015 1 625015

Step 3.Step 3. Calculate the value of the test statistic

Triple A Construction Step 4.Step 4.

Reject the null hypothesis if the test statistic is greater than the F-value in Appendix D

df1 = k = 1 df2 = n – k – 1 = 6 – 1 – 1 = 4

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df2 = n – k – 1 = 6 – 1 – 1 = 4

The value of F associated with a 5% level of significance and with degrees of freedom 1 and 4 is found in Appendix D

F0.05,1,4 = 7.71 Fcalculated = 9.09 Reject H0 because 9.09 > 7.71

Triple A Construction

We can conclude there is a statistically significant relationship between X and Y

The r2 value of 0.69 means about 69% of the variability in

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F = 7.71

9.09Figure 4.5

about 69% of the variability in sales ( Y) is explained by local payroll ( X)

Triple A Construction

The F-test determines whether or not there is a relationship between the variables

r2 (coefficient of determination) is the best measure of the strength of the prediction

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measure of the strength of the prediction relationship between the X and Y variables Values closer to 1 indicate a strong prediction

relationship Good regression models have a low

significance level for the F-test and high r2

value.

Analysis of Variance (ANOVA) Table

When software is used to develop a regression model, an ANOVA table is typically created that shows the observed significance level ( p-value) for the calculated F value

This can be compared to the level of significance (αααα) to make a decision

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Regression k SSR MSR = SSR/k MSR/MSE P(F > MSR/MSE)

Residual n - k - 1 SSE MSE = SSE/(n - k - 1)

Total n - 1 SST

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Because this probability is less than 0.05, we reject the null hypothesis of no linear relationshi p and conclude there is a linear relationship between X and Y

Program 4.1D (partial)

P(F > 9.0909) = 0.0394

Multiple Regression Analysis

Multiple regression modelsMultiple regression models are extensions to the simple linear model and allow the creation of models with several independent variables

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Y = dependent variable (response variable) Xi = ith independent variable (predictor or explanatory

variable) ββββ0 = intercept (value of Y when all Xi = 0) ββββI = coefficient of the ith independent variable k = number of independent variables εεεε = random error

Multiple Regression Analysis

To estimate these values, samples are taken and the following equation is developed

kk XbXbXbbY ++++++++++++++++==== ...ˆ 22110

where = predicted value of Y

b0 = sample intercept (and is an estimate of ββββ0) bi = sample coefficient of the ith variable (and is

an estimate of ββββi)

Jenny Wilson Realty

Jenny Wilson wants to develop a model to determine the suggested listing price for houses based on the size and age of the house

22110 ˆ XbXbbY ++++++++====

where = predicted value of dependent variable (selling pri ce)

b0 = Y intercept X1 and X2 = value of the two independent variables (square

footage and age) respectively b1 and b2 = slopes for X1 and X2 respectively

Y

She selects a few samples of the houses sold recently and records the data shown in Table 4.5

She also saves information on house condition to be used later

Jenny Wilson Realty SELLING PRICE ($)

SQUARE FOOTAGE AGE CONDITION

95,000 1,926 30 Good

119,000 2,069 40 Excellent

124,800 1,720 30 Excellent

135,000 1,396 15 Good

142,000 1,706 32 Mint

Jenny Wilson Realty

0021788.0

Evaluating Multiple Regression Models

Evaluation is similar to simple linear regression models The p-value for the F-test and r2 are

interpreted the same

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The hypothesis is different because there is more than one independent variable The F-test is investigating whether all

the coefficients are equal to 0 If the F-test is significant, it does not

mean all independent variables are significant

Evaluating Multiple Regression Models

To determine which independent variables are significant, tests are performed for each variable

010 ====ββββ:H

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010 ====ββββ:H 011 ≠≠≠≠ββββ:H

The test statistic is calculated and if the p-value is lower than the level of significance ( αααα), the null hypothesis is rejected

Jenny Wilson Realty

The model is statistically significant The p-value for the F-test is 0.002 r2 = 0.6719 so the model explains about 67% of

the variation in selling price ( Y) But the F-test is for the entire model and we can’t

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But the F-test is for the entire model and we can’t tell if one or both of the independent variables ar e significant

By calculating the p-value of each variable, we can assess the significance of the individual variables

Since the p-value for X1 (square footage) and X2 (age) are both less than the significance level of 0.05, both null hypotheses can be rejected

Binary or Dummy Variables

BinaryBinary (or dummydummy or indicatorindicator) variables are special variables created for qualitative data

A binary variable is assigned a value of 1 if a particular qualitative condition is met and

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a particular qualitative condition is met and a value of 0 otherwise

Adding binary variables may increase the accuracy of the regression model

The number of binary variables must be one less than the number of categories of the qualitative variable

Jenny Wilson Realty

Jenny believes a better model can be developed if she includes information about the condition of the property

X3 = 1 if house is in excellent condition = 0 otherwise

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= 0 otherwise X4 = 1 if house is in mint (perfect) condition

= 0 otherwise

Two binary variables are used to describe the three categories of condition

No variable is needed for “good” condition since if both X3 = 0 and X4 = 0, the house must be in good condition

Jenny Wilson Realty

Model explains about 89.8% of the variation in selling price

F-value

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Program 4.3 – The two additional dummy variables result in higher r2 and smaller significance value.

F-value indicates significance

Model Building

The best model is a statistically significant model with a high r2 and few variables

As more variables are added to the model, the r2-value usually increases However more variables does not

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However more variables does not necessarily mean better model

For this reason, the adjusted adjusted rr22 value…